As "Son of God", Jesus Christ shares the same qualities of being with the Father and the Spirit. Such qualities include omniscience, omnipotence, omnipresence, eternality and all the attendant attributes that makes Him truly God along with the Father and the Spirit.
As the "Word", the focus of John's opening verses is to make known to us how Jesus as "The Son of God" functions within the Godhead to express in an intelligible way the glory of the Father. Other biblical passages such as Hebrews 1:1-3 describe this activity. John 1:18b captures this profound truth with amazing brevity: "the only begotten God who is in the bosom of the Father, He has explained Him." This is what is meant by the title "Son of Son".
Perhaps expressed another way - the title "God the Son" serves to express the "Godness" or "Deity" of the Son as He has existed from all eternity. Thus, when Jesus explains Himself in John 8:58 by the phrase: "before Abraham was, I am", He is expressing how He is truly Divine.
We find these words in the midst of John's prologue, John 1:14: "And the Word became flesh, and dwelt among us, and we saw His glory, glory as of the only begotten from the Father, full of grace and truth." Unless the Son has willingly incarnated Himself in human flesh, there would had been no salvation nor capability of Personally knowing nor interacting with Him.
of God made flesh and rejoice in our great salvation through Jesus Christ our Lord. With the Father and the Holy Spirit, the Son created all things, sustains all things, and makes all things new. Truly God, He became truly man, two natures in one person." The link for the full Ligonier statement can be found at: http://christologystatement.com/